Abstract. The mechanical and thermodynamic properties (such as the
nearest neighbor distance, the molar volume, the adiabatic and isothermal
compressibilities, the thermal expansion coefficient and the heat capacities
at constant volume and at constant pressure) of molecular cryocrystals of
many atoms with a face-centered cubic structure such as α-CO2, α-N2O, at
various temperatures and at pressures up to 10 GPa are investigated using the
statistical moment method (SMM) in statistical mechanics and compared with the
experimental data.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 119-125
This paper is available online at
MECHANICAL AND THERMODYNAMIC PROPERTIES
OF CO2 AND N2OMOLECULAR CRYOCRYSTALS UNDER PRESSURE
Nguyen Quang Hoc1, Bui Duc Tinh1 and Nguyen Duc Hien2
1Faculty of Physics, Hanoi National University of Education
2Mac Dinh Chi Secondary School, Chu Pah District, Gia Lai Province
Abstract. The mechanical and thermodynamic properties (such as the
nearest neighbor distance, the molar volume, the adiabatic and isothermal
compressibilities, the thermal expansion coefficient and the heat capacities
at constant volume and at constant pressure) of molecular cryocrystals of
many atoms with a face-centered cubic structure such as -CO2, -N2O, at
various temperatures and at pressures up to 10 GPa are investigated using the
statistical moment method (SMM) in statistical mechanics and compared with the
experimental data.
Keywords:Molecular cryocrystal, statistical moment method.
1. Introduction
Molecular crystals are characterized by strong intramolecular forces and much
weaker intermolecular forces. High-pressure spectroscopic studies provide useful data for
refining the various model potentials which are used to predict of the physical properties
of such systems as well as for the formation of various crystalline phases.
CO2 is an important volatile component of the earth as well as other planets in
the solar system. Its high-pressure behavior is therefore of fundamental importance in
planetary science. CO2 is one of the model systems involving the π bonding and the
hybridization properties of the carbon atom, which are strongly affected by high pressure
conditions.
Pressure-induced transitions from molecular to nonmolecular CO2 crystals
are systematically investigated using first-principle lattice dynamics calculation.
Geometrically, likely transition pathways are derived from the dynamical instability of
the molecular crystals under high pressures.
The phase diagram of CO2 consists of 5 phases. CO2-I phase or phase α, known as
dry ice) has the face-centered cubic Pa3 structure. CO2-II has the P42/mnm symmetry.
Received August 20, 2014. Accepted October 1, 2014.
Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn
119
Nguyen Quang Hoc, Bui Duc Tinh and Nguyen Duc Hien
CO2-III has the orthorhombic Cmca symmetry. CO2-IV has Pbcn symmetry. CO2-V is
the polymeric phase of a tridymite-like structure. In [1], Bonev et al. performed a series
of first-principle calculations, including full structural optimizations, phonon spectra
and free energies, in order to study the stability and properties of the phases proposed
experimentally up to 50 GPa and 1500 K. The DFT calculations were carried out within
the Perdew-Burke-Ernzerhof generalized gradient approximation (CGA) [2] using the
ABINIT code which implements plane-wave basis sets.
Le Sar et al. [3] presented an ab initio method, based on the modified Gordon-Kim
(MGK) electron-gas model which worked well in calculating the structure and properties
of molecular crystals.
A constant pressure Monte Carlo formalism, lattice dynamics and classical
perturbation theory are used to calculate the thermal expansion, the pressure-volume
relation at room temperature, the temperature dependence of zone center libron
frequencies and the pressure dependence of the three vibron modes of vibration in solid
CO2 at pressures 0 ≤ p ≤ 16 GPa and temperatures 0 ≤ T ≤ 300 K [4].
Properties of solid N2O at pressures p ≤ 15Gpa and at T = 0 and 300 K have been
calculated using energy optimization and Monte Carlo methods in an (N, p, T ) ensemble
with periodic, deformable boundary conditions and lattice dynamics. α-N2O is consistent
with the known low-pressure low-temperature ordered cubic form, space group Pa3, up
to 4.8 GPa where transition to a new solid occurs [5].
Cryocrystals N2O and CO2 are ideal systems on which to have a study of the
influence of quantum effects on condensed matter. There has been considerable interest
in structural and thermodynamic properties of these crystals under temperature and
pressure and in line with this general interest and encouraged by the essential success
of our calculations, we tried to consider the mechanical and thermodynamic properties
of cryocrystals of many atoms with face-centered cubic structure such as α-N2O, α-CO2
at various temperatures and pressures up to 10 GPa. Heat capacities at constant volume
for these crystals are studied by combining the SMM and the self-consistent field method
taking into account the lattice vibration and the molecular rotational motion [6].
2. Content
2.1. Mechanical and thermodynamic properties of cryocrystals α-CO2
and α-N2O at pressure p = 0
It is known that the interaction potential between two atoms in α phase of molecular
cryocrystals of N2 type such as solids N2, CO, CO2 and N2O is usually used in the form
of the Lennard-Jones pair potential
ϕ(r) = 4ε
[(σ
r
)12
−
(σ
r
)6]
(2.1)
where σ is the distance in which ϕ(r) = 0 and ε is the depth of the potential well.
120
Mechanical and thermodynamic properties of CO2 and N2O molecular cryocrystals...
The values of the parameters ε, σ are determined from the following experimental
data. ε/kB = 218.82K, σ = 3.829.10−10m for β-CO2 and ε/kB = 235.48K, σ =
3.802.10−10m for α-N2O [8]. Therefore, using the coordinate sphere method and the
results in [7], we obtain the values of parameters for α-CO2 and α-N2O as follows:
k =
4ε
a2
(σ
a
)6 [
265.298
(σ
a
)6
− 64.01
]
,
γ =
16ε
a4
(σ
a
)6 [
4410.797
(σ
a
)6
− 346.172
]
γ1 =
4ε
a4
(σ
a
)6 [
803.555
(σ
a
)6
− 40.547
]
,
γ2 =
4ε
a4
(σ
a
)6 [
3607.242
(σ
a
)6
− 305.625
]
. (2.2)
where a is the nearest neighbor distance at temperature T. Our calculated results for
the nearest neighbor distance a, the adiabatic and isothermal compressibilities χT , χS,
the thermal expansion coefficient β and the heat capacities at constant volume and
constant pressure CV , Cp of α-CO2 and α-N2O at different temperatures and pressure
p = 0 are shown in [7]. In general, our calculations are in qualitative agreement with
experimental results.
2.2. Mechanical and thermodynamic properties of cryocrystals α-CO2
and α-N2O under pressure
In order to determine thermodynamic quantities at various pressures, we must find
the nearest neighbor distances. The equation for calculating the nearest neighbor distances
at pressure P and at temperature T has the form [7]
y2 = 1.1948 +
[
0.1717 + 0.0862
θ
ε
xcthx
]
y4 − 0.0087pσ
3
ε
y5
− 0, 0019θ
ε
xcthxy6 + 0.0021
pσ3
ε
y7. (2.3)
where y =
(
a
σ
)3
, θ = kBT (kB is the Boltzmann constant), x = ~ω2θ . This is a nonlinear
equation and therefore, it has only an approximate solution. From that, the equation
for calculating the nearest neighbor distances at pressure p and at temperature 0 K has
the form
y2 = 1.1948 + 0.1717y4 − 0..0087pσ
3
ε
y5 + 0.0021
pσ3
ε
y7. (2.4)
After finding the solution a (P, 0 K) from (2.4), we can calculate a (P, T) and other
thermodynamic quantities. This means is applied to crystal at low pressures. For crystal
at high pressures, we must find the solution using (2.4).
121
Nguyen Quang Hoc, Bui Duc Tinh and Nguyen Duc Hien
For example in the case of α-CO2 at p = 0.5 kbar, T = 0 K, (2.4) becomes
y2 = 1.1948 + 0.17y4 − 0.00807y5 + 0.082y7. (2.5)
The solution of this equation is y = 1.281967, i.e. the nearest neighbor distance
under the condition p = 0.5 kbar, T = 0K takes a value a = 4.1578.10−10 m. At
temperature 0 K and pressure p, the parameters of α-CO2 and α-N2O are summarized
in Table 1. Our calculated results for thermodynamic quantities of α-CO2 and α-N2O at
different temperatures and pressures up to 10 GPa are shown in Figures 1-11. According
to the experimental data, α-CO2 exists in the pressure range of 0 to 12 GPa and in the
temperature range of 0 to 120 K and α-N2O exists in the pressure range of 0 to 4.8 GPa
and in the temperature range of 0 to 130 K. Our numerical results are carried out in these
ranges of temperature and pressure. We have only the experimental data for the phase
diagram and the molar volume of α-CO2 and α-N2O under pressure. The dependence of
thermodynamic quantities on temperature for α-CO2 and α-N2O crystals at pressure p
= 0 and at preesure p ̸= 0 have same behaviour. Our results would be more consistent
with experiments if we take molecular rotation and intermolecular motion into account.
Our obtained results can be enlarged to cases in higher pressures. Our calculated results
for molecular crystals α-CO2 and α-N2O show that at same pressure, when temperature
increases heat capacities CV and Cp increase. At same temperature, when pressure
increases the heat capacities CV and Cp decrease. In the interval of pressure shown in
figures, when temperature T < 20 K, heat capacities CV and Cp approximately are equal
to zero. At mentioned pressures, in the range from 50 K to 110 K, heat capacities CV
and Cp increase strongly. At same temperature, when pressure increases the value of heat
capacity CV comes to the value of heat capacity CP .
Table 1. Parameters of −CO2 and −N2O at p = 0.5 kbar, 1 kbar and T = 0 K
Crystal p, kbar k, J/m2 !,1013s1
,
1021J/m2
1,
1021J/m2
2,
1021J/m2
a0,
1010m
CO2
0.5 4.1687 2.2869 2.3117 0.1108 0.4671 4.1578
1 4.4444 2.3613 2.4559 0.1176 0.4964 4.1430
N2O
0.5 4.5225 2.3649 2.5446 0.1220 0.5141 4.1299
1 4.7967 2.4356 2.6900 0.1288 0.5437 4.1163
122
Mechanical and thermodynamic properties of CO2 and N2O molecular cryocrystals...
Figure 7. Graphs of CV (T ), Cp(T ) for α−CO2 at p = 0, p = 0.5 kbar and p = 1 kbar
123
Nguyen Quang Hoc, Bui Duc Tinh and Nguyen Duc Hien
Figure 8. Graphs of CV (T ), Cp(T ) for α−N2O at p = 0, p = 0.5 kbar and p = 1 kbar
Figure 11. Dependence of relative change of molar volume on pressure at temperature
77 K for α−CO2 from our calculated result (SMM) and experiments (EXPT) [9, 10]
124
Mechanical and thermodynamic properties of CO2 and N2O molecular cryocrystals...
3. Conclusion
In this paper, we calculate thermodynamic properties such as the nearest
neighbor distance, the isothermal and adiabatic compressibilities, the thermal expansion
coefficient, the heat capacities at constant volume and at constant pressure of cryocrystals
α-CO2 and α-N2O with fcc structure at low pressures p = 0, 0.5 and 1 kbar and the
nearest neighbor distance of cryocrystals CO2 and N2Owith fcc structure at high pressures
p = 2, 6 and 10 GPa at different temperatures. Our calculated result for the relative
change of molar volume versus pressure at temperature 77 K for α-CO2 is compared
with the experimental data. In comparison with the experimental data, for some values
of quantities such as the nearest neighbur distance, the thermal expansion coefficient, our
obtained results are relatively good but for quantities such as the adiabatic and isothermal
compressibilities, the heat capacities at constant volume and at constant pressure our
obtained results only agree in the order of magnitude. The dependence of thermodynamic
quantities on temperature for α-CO2 and α-N2O under pressure is in physical agreement
with that at zero pressure. Our results will be more consistent with experiments by taking
account of molecular rotation and intermolecular motion. Our obtained results can be
enlarged to cases in higher pressures.
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